Is it a proper method to use span tag inside H1 tag?
<h1 style="display:inline;">Bold text goes here</h1> 
<span style="display:inline;">normal text goes here</span>
Think in above lines - Worked for me - use display:inline prop
Yes you can. It can be used to format a part of a h1 block:
<h1>Page <span class="highlight">Title</span></h1>
If the style applies to the entire h1 block, I do this:
<h1 class="highlight">Page Title</h1>
Yes, we can use span tag with header tags and there is nothing wrong in it. Indeed this is widely used for styling header tags, specially for coloring a particular word or letter.
Yes you can.
HTML4 has this to say:
<!ENTITY % heading "H1|H2|H3|H4|H5|H6">
<!--
  There are six levels of headings from H1 (the most important)
  to H6 (the least important).
-->
<!ELEMENT (%heading;)  - - (%inline;)* -- heading -->
And %inline; is:
<!ENTITY % inline "#PCDATA | %fontstyle; | %phrase; | %special; | %formctrl;">
And %special; includes <span>.
The current HTML has this to say:
Content contents
Phrasing content
And Phrasing content includes <span>.
Yes, we can use span tag with header tags and there is nothing wrong in it. Indeed this is widely used for styling header tags, specially for coloring a particular word or letter.
Yes, it's typically fine to use a span inside of an h1.  span is an inline element, so it's typically okay to use it inside anything (that allows elements inside it!)
And there's not really a cleaner way to do it sometimes, say if you want to style only part of the h1.
On the other hand... don't do it if it's not necessary, as it is a little ugly : )